QEDMonthly Mini-Issue
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AdHoc Problems
November 30, 2014
Introduction
There are certain problems on the AMC’s that require almost no mathematical knowledge beyond
the scope of simple manipulation of fractions and ratios, known as AdHoc Problems. These concepts
and topics have been introduced to you since you were in elementary school and therefore should
be as familiar to you as the palm of your hand. However, more often than not, many students get
these questions wrong due to silly mistakes, wrong interpretations of the problem, or simply because
they don’t understand the technique to solve the problem. The most common of these problems is
Time and Distance Problems, which we will show how to solve as well as introduce other AdHoc
problems. The key step to solving any of these questions is to correctly translate what is given in the
problem, into a set of equations that one can solve algebraically.
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Formulas and Prior Knowledge
The amount of knowledge needed for these types of problems are extremely minimal. In fact, it’s one
equation:
Distance = Rate · T ime
Because of this minimal knowledge, these types of problems are extremely popular on the AMC’s.
However, don’t let this simple equation fool you as sometimes these problems are extremely tricky
and cause problem solvers to completely miss the question.
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Examples to Learn
Example 1: Brenda and Sally run in opposite directions on a circular track, starting at diametrically
opposite points. Each girl runs at a constant speed. They first meet after Brenda has run 100 meters.
They next meet after Sally has run 150 meters past their first meeting point. What is the length of
the track in meters?
(A) 250
(B) 300
(C) 350
(D) 400
(E) 500
Solution: In any of these types of equations, always set variables to get a clear idea of what’s going on in the problem. Let Vb be the speed of Brenda, Vs the speed of Sally, t1 the time of the first
run, t2 the time of the second run, and D be the length of the track. Now we can set up the equations
for the first time they meet:
Vb · t1 = 100
(1)
(Vb + Vs ) · t1 = 1/2 · D
(2)
The second equation comes from the fact that they start at diametrically opposite points which means
when they cross for the first time they will run half the track. Now we will set up the equations for
the second time they meet.
(Vb + Vs ) · t2 = D
(3)
Vs · t2 = 150
(4)
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QEDMonthly Mini-Issue
AdHoc Problems
November 30, 2014
The most important observation here is that t2 = 2 · t1 which comes from dividing Equation (3)
by Equation (2). Now we have all the equations we need and we simply solve the system of linear
equations. Substituting this observation into Equation (1) gives us that Vb · t2 = 200. Substituting
this equation and Equation (4) into Equation (3) gives us that D = 350 which gives us the answer is
(C) .
Example 2: Cassandra sets her watch to the correct time at noon. At the actual time of 1:00
PM, she notices that he watch reads 12:57 and 36 seconds. Assume that her watch loses time at a
constant rate. What will be the actual time when her watch first reads 10:00 PM.
(A) 10:22 PM and 24 seconds
(B) 10:24
(C) 10:25
(D) 10:27
(E) 10:30
Solution: Since the clock loses time at a constant rate we simply need to know the ratio of of the
speed of the watch to the speed of the actual time. Say that a minute is our unit measurement. So
by the given information problem, the actual time has ”traveled” 60 minutes while the watch has
288
”traveled” 57.6 minutes which is the same as
minutes. Since both actual time and the watch
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”traveled” during the same interval of time the ratio of the speed of the actual time to the speed of
25
60
the watch is simply: 288 = . Now if the watch has traveled 600 minutes (10 hours), the actual time
24
5
25
should be
· 600 = 625, which corresponds to 10:25 and thus the answer is (C) .
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Example 3: Suppose hops, skips, and jumps are specific units of length. We know that b hops
equal c skips, d jumps equal e hops, and f jumps equal g meters. How many skips are equal to one
meter?
(A)
bdg
cef
(B)
cdf
beg
(C)
cdg
bef
(D)
cef
bdg
(E)
ceg
bdf
Solution: This problem is a simple case of dimensional analysis. The following line of calculations
immediately solves the problem:
1 meters ·
f jumps e hops c skips
cef
·
·
=
skips
g meters d jumps b hops
bdg
which gives the answer (D) .
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Problem Set
1. David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour,
but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by
15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many
miles is the airport from his home?
(A) 140
(B) 175
(C) 210
(D) 245
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(E) 280
QEDMonthly Mini-Issue
AdHoc Problems
November 30, 2014
2. Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will
d cows give in e days?
(A)
bde
ac
(B)
ac
bde
(C)
adbe
c
(D)
bcde
a
(E)
abc
de
3. Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the
stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time
of 3 hours including the stop. Which equation could be used to solve for the time t in hours
that she drove before her stop?
(A) 80t + 100(8/3 − t) = 250
(B) 80t = 250
(C) 100t = 250
(E) 80(8/3 − t) + 100t = 250
(D) 90t = 250
4. Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages 40 miles per hour, he arrives at his workplace three minutes late. When he averages 60
miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should
Mr. Bird drive to arrive at his workplace precisely on time?
(A) 45
(B) 48
(C) 50
(D) 55
(E) 58
5. Jack and Jill run 10 kilometers. They start at the same place, run 5 kilometers up a hill, and
return to the starting point by the same route. Jack has a 10-minute head start and runs at a
rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill.
How far from the top of the hill are they when they pass going in opposite directions?
(A)
5
4
(B)
35
27
(C)
27
20
(D)
7
3
(E)
28
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6. It takes Clea 60 seconds to walk down an escalator when it is not operating and only 24 seconds
to walk down the escalator when it is operating. How many seconds does it take Clea to ride
down the operating escalator when she just stands on it?
(A) 36
(B) 40
(C) 42
(D) 48
(E) 52
7. Andrea and Lauren are 20 kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of 1
kilometer per minute. After 5 minutes, Andrea stops biking because of a flat tire and waits for
Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
(A) 20
(B) 30
(C) 55
(D) 65
(E) 80
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QEDMonthly Mini-Issue
AdHoc Problems
November 30, 2014
8. Three runners start running simultaneously from the same point on a 500-meter circular track.
They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0
meters per second. The runners stop once they are all together again somewhere on the circular
course. How many seconds do the runners run?
(A) 1000
(B) 1250
(C) 2500
(D) 5000
(E) 10000
9. In a h-meter race, Sam is exactly d meters ahead of Walt when Sam finishes the race. The next
time they race, Sam sportingly starts d meters behind Walt, who is at the original starting line.
Both runners run at the same constant speed as they did in the first race. how many meters
ahead is the winner of the second race when the winner crosses the finish line?
(A)
d
h
(B) 0
(C)
d2
h
(D)
h2
d
(E)
d2
h−d
10. Paula the painter and her two helpers each paint at constant, but different, rates. They always
start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday
the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn’t
there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula
worked by herself and finished the house by working until 7:12 PM. How long, in minutes, was
each day’s lunch break?
(A) 30
(B) 36
(C) 42
(D) 48
(E) 60
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